Factoring a Sum or Difference of Whole Numbers

We can have sums or differences of whole numbers; for example (26 + 65) or (48 − 16).

For factoring such sums or differences of whole numbers:

We write the whole numbers as products of their prime factors.

Then we factor out the greaterst common factors (gcf) from those numbers

We factor out any given common factor, if required, from such sums or differences of whole numbers.

Example:

Factor out the gcf from the sum (28 + 63)

Solution

The prime factorization of 28 is 28 = 4 × 7

The prime factorization of 63 is 63 = 9 × 7

So the greatest common factor or gcf of 28 and 63 is 7

So (28 + 63) = (4 × 7 + 9 × 7) = 7(4 + 9)

Factor out the gcf from the sum of whole numbers (26 + 91)

Solution

Step 1:

26 = 2 × 13

91 = 7 × 13

Step 2:

The gcf of 26 and 91 is 13. So factoring out the greatest common factor 13

(26 + 91) = (2 × 13 + 7 × 13)= 13(2 + 7)

Factor out 6 from the difference of whole numbers (108 − 84)

Solution

Step 1:

84 = 2 × 2 × 3 × 7 = 6 × 14

108 = 2 × 2 × 3 × 3 × 3 = 6 × 18

Step 2:

So factoring out 6 from the difference of the given numbers

(108 − 84) = (6 × 18 − 6 × 14) = 6(18 − 14)